These are perfectly valid questions about intuitionistic, or constructive, logic.

Scubasteve123, when you are asking questions with narrow specialization, it may make sense to briefly describe the context; otherwise, people not familiar with the subject don't know whether it is nonsense or just something they don't know.

1) Premise: There exists an x Fx----> for all x Gx
Conclusion: for all x(Fx--->Gx)

I assume parentheses are as follows: implies . This is true. Fix an and assume . From this we get , and, therefore, and so .

ie: you can get a proof of p from a proof of <not><not>p (double negation p)

It's the other way around. From p one can prove (not (not p)) but in general not conversely.

Usually, is taken as an abbreviation for ( denotes falsehood). So, given and an assumption one gets by Modus Ponens. Closing the assumption one gets , i.e., .

An intriguing thing is that one does not use the properties of in the derivation above. I.e., from one can prove for any formula .

One can also remove double negations, but only if the remaining formula itself starts with a negation: . In general, there is no difference between classical and intuitionistic logic for formulas that start with a negation: classically implies \ intuitionistically (Glivenko's theorem).